process and the residuals are realizations of the error process. Each residual e, is an observation on the random variable e(; the residuals will depend on the values obtained for the parameter estimates a and p.

How should these estimates be obtained? It is logical to choose a method of estimation that in some way minimizes the residuals, since then the predicted values of the dependent variable will be closer to the observed values. Choosing estimates to minimize the sum of the residuals will not work, because large positive residuals would cancel large negative residuals. The sum of the absolute residuals could be minimized, but the mathematical properties of the estimators are much nicer if we minimize the sum of the squared residuals. This is the ordinary least squares (OLS) criterion.

If coefficients are chosen to minimize the sum of squared residuals, the OLS estimators for coefficients in the simple linear model (A. 1.2) are given by:

5 = (1,- - )/(1-1)2, (A.1.4)

t I t

a=Y-bX, (A. 1.5)

where X, Y denote the sample means (that is, the arithmetic averages) of X and Y. Thus OLS estimate of P uses an equally weighted variance estimate and an equally weighted covariance estimate, with the number of data points in the average being the number of data points used for the regression and without a zero mean assumption (§3.1):

fe = est.cov(X„ Yt)/est.V(Xt). (A. 1.6)

In financial markets plenty of data are often available. Therefore one should choose an estimation method that gives consistent estimators (§A.1.3). OLS is widely used because it gives consistent estimators in fairly general circumstances. However, in small samples OLS will only give the best estimates under certain conditions, and an alternative method of estimation should be used if residual diagnostic tests indicate that these conditions do not hold (§A.3.3).

A. 1.2 Multivariate Models

The general linear statistical model is

Y, = R,*,, + 1}2X2, + ... + P,X„ + e„ (A. 1.7)

and assuming the model contains a constant term then Xu - \ for all t= 1, . . ., T. There will be T equations in the unknown parameters p,, P2, . . ., Pb one equation for each date in the data. Model estimation (or fitting the model) involves choosing a method for solving these equations and then using data to obtain estimates P,, P2, . . ., Pfc of the model parameters.

The fitted model may be used to predict the values of Y corresponding to given values of the independent variables. At a particular time t each set of values for Xu X2, . ., Xk determines a predicted value of Y given by

Y, = p,*,, + P2X2, + . . . + $kXkt.

The difference between the actual and predicted value of Y is the residual e„ therefore

Y, = &XU + p2X2r + . . . + faXkt + et. (A. 1.8)

Care should be taken to distinguish the model (A. 1.8), where the values estimated for the coefficients and residuals depend on the data employed, from the theoretical model (A. 1.7).

The general linear model may also be written using matrix notation. Let the column vector of dependent variable data be y = (Tb Y2, . ., YT) and arrange the data on the independent variables into a matrix X, so that the jth column of X is the data on A}. The first column of X will be a column of Is if there is a constant term in the model. Denote by p = (p,, p2, . . ., p) the vector of true parameters and £ = (e,, e2, . . ., eT) the vector of error terms. Then the matrix form of the general linear model (A. 1.7) is

= + . (A.1.9)

The matrix form of the equations for the OLS estimators of p in the general linear model (A.1.9) is1

b = (XX)-Xy. (A.1.10)

Since X is a T xk matrix, we need to calculate the inverse of the symmetric xk matrix XX, and multiply it (on the left) by the x 1 vector Xy to get the x 1 vector of OLS estimates. Here is a simple example of the use of formula (A. 1.10) that can be replicated using a hand calculator. To estimate a deterministic trend in a time series, consider the regression of Y on a constant and time:

Yt = a + / + ,.

Suppose there are five observations on Y: , = 10, Y2 = 13, Y3 = 19, Y4 = 18 and Y5 = 20, and so = (10, 13, 19, 18, 20). The matrix of data on the explanatory variables X will be a 5 x 2 matrix whose first column contains Is (for the constant) and whose second column contains 1, 2, 3, 4, 5 (for the time trend). Thus

]Note that the OLS estimators of p are generally denoted b = (6,, 62, . . ., bk)\ the hat notation being used to denote arbitrary estimators. Note also that in the case = 2 equations (A.1.10) may be written in the summation form given for the OLS equations in (A.1.4) and (A.1.5).

Figure A.2 Scatter plot with OLS regression line and error process.

x,= (,j H), (X*,-= ( *»

Applying (A. 1.10), we obtain the vector b = (a, b) of OLS estimates of P = (a, P) as

/ 55/50 -15/50W 80 \ /8.5\ -15/50 5/50 265) ~ \2.5)-

Therefore the OLS fitted model is Y = 8.5 + 2.5/ as shown in Figure A.2.

A. 1.3 Properties of OLS Estimators

When applying OLS, or any other estimation method, it should be understood that there is only one theoretical model, but there may be any number of estimated models corresponding to different data and/or different estimation methods. There is only one OLS estimator for each coefficient in the model, but there will be many different OLS estimates of any coefficient, depending on the data used.

To illustrate this point, suppose the model (A. 1.2) is estimated for the Brazilian data illustrated in Figure A.l. Although there is only one model, different values for the coefficients are obtained depending on whether the data used are weekly or daily. Using the OLS formulae (A. 1.4) and (A.1.5) for the same period from 1 August 1994 to 30 December 1997, one obtains

Y= -9.3 x 10"4 + 1.2633X

on weekly returns data, and

f = -2.5 x 10 4 + 1.2111